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{{redirect|Divisible|divisibility of groups|Divisible group}} | |||
{{about||the second operand of a division|Division (mathematics)|divisors in algebraic geometry|Divisor (algebraic geometry)|divisibility in the ring theory|Divisibility (ring theory)}} | |||
[[File:Cuisenaire ten.JPG|thumb|The divisors of 10 illustrated with [[Cuisenaire rods]]: 1, 2, 5, and 10]] | |||
In [[mathematics]] a '''divisor''' of an [[integer]] <math>n</math>, also called a '''factor''' of <math>n</math>, is an integer that can be multiplied by some other integer to produce <math>n</math>. | |||
==Definition== | |||
Two versions of the definition of a divisor are commonplace: | |||
* For integers <math>m</math> and <math>n</math>, it is said that <math>m</math> '''divides''' <math>n</math>, <math>m</math> is a '''divisor''' of <math>n</math>, or <math>n</math> is a '''multiple''' of <math>m</math>, and this is written as | |||
*:<math>m \mid n</math>, | |||
:if there exists an integer <math>k</math> such that <math>mk = n</math>.<ref>for instance, {{harvnb|Sims|1984|loc=p. 42}} or {{harvnb|Durbin|1992|loc=p. 61}}</ref> Under this definition, the statement <math>0 \mid 0</math> holds. | |||
* As before, but with the additional constraint <math>m \neq 0</math>.<ref>{{harvnb|Herstein|1986|loc=p. 26}}</ref> Under this definition, the statement <math>0 \mid 0</math> does not hold. | |||
In the remainder of this article, which definition is applied is indicated where this is significant. | |||
==General== | |||
Divisors can be [[negative number|negative]] as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned. | |||
1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself.<ref>This statement either requires 0|0 or needs to be restricted to nonzero integers.</ref> Every integer is a divisor of 0.<ref>This statement either requires 0|0 or needs to be restricted to nonzero integers.</ref> Integers divisible by 2 are called [[even and odd numbers|even]], and numbers not divisible by 2 are called [[even and odd numbers|odd]]. | |||
1, −1, ''n'' and −''n'' are known as the '''trivial divisors''' of ''n''. A divisor of ''n'' that is not a trivial divisor is known as a '''non-trivial divisor'''. A non-zero integer with at least one non-trivial divisor is known as a [[composite number]], while the [[Unit (ring theory)|units]] −1 and 1 and [[prime number]]s have no non-trivial divisors. | |||
There are [[divisibility rule]]s which allow one to recognize certain divisors of a number from the number's digits. | |||
The [[Generalization (logic)|generalization]] can be said to be the concept of ''divisibility'' in any [[integral domain]]. | |||
==Examples== | |||
*7 is a divisor of 42 because <math>7 \times 6 = 42</math>, so we can say <math>7 \mid 42</math>. It can also be said that 42 is '''divisible''' by 7, 42 is a [[multiple (mathematics)|multiple]] of 7, 7 '''divides''' 42, or 7 is a '''factor''' of 42. | |||
*The non-trivial divisors of 6 are 2, −2, 3, −3. | |||
*The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. | |||
*<math>5 \mid 0</math>, because <math>5 \times 0 = 0</math>. | |||
*The [[Set (mathematics)|set]] of all positive divisors of 60, <math>A = \{ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 \}</math>, [[Partially ordered set|partially ordered]] by divisibility, has the [[Hasse diagram]]: | |||
[[File:Lattice of the divisibility of 60; factors.svg|center|350px]] | |||
==Further notions and facts== | |||
There are some elementary rules: | |||
* If <math>a \mid b</math> and <math>b \mid c</math>, then <math>a \mid c</math>. This is the [[transitive relation]]. | |||
* If <math>a \mid b</math> and <math>b \mid a</math>, then <math>a = b</math> or <math>a = -b</math>. | |||
* If <math>a \mid b</math> and <math>a \mid c</math>, then <math> a \mid (b + c)</math> holds, as does <math> a \mid (b - c)</math>.<ref><math>a \mid b,\, a \mid c \Rightarrow b=ja,\, c=ka \Rightarrow b+c=(j+k)a \Rightarrow a \mid (b+c)</math>. Similarly, <math>a \mid b,\, a \mid c \Rightarrow b=ja,\, c=ka \Rightarrow b-c=(j-k)a \Rightarrow a \mid (b-c)</math></ref> However, if <math>a \mid b</math> and <math>c \mid b</math>, then <math>(a + c) \mid b</math> does ''not'' always hold (e.g. <math>2\mid6</math> and <math>3 \mid 6</math> but 5 does not divide 6). | |||
If <math>a \mid bc</math>, and [[greatest common divisor|gcd]]<math>(a, b) = 1</math>, then <math>a \mid c</math>. This is called [[Euclid's lemma]]. | |||
If <math>p</math> is a prime number and <math>p \mid ab</math> then <math>p \mid a</math> or <math>p \mid b</math>. | |||
A positive divisor of <math>n</math> which is different from <math>n</math> is called a '''proper divisor''' or an '''aliquot part''' of <math>n</math>. A number that does not evenly divide <math>n</math> but leaves a remainder is called an '''aliquant part''' of <math>n</math>. | |||
An integer <math>n > 1</math> whose only proper divisor is 1 is called a [[prime number]]. Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself. | |||
Any positive divisor of <math>n</math> is a product of [[prime factor|prime divisor]]s of <math>n</math> raised to some power. This is a consequence of the [[fundamental theorem of arithmetic]]. | |||
A number <math>n</math> is said to be [[perfect number|perfect]] if it equals the sum of its proper divisors, [[deficient number|deficient]] if the sum of its proper divisors is less than <math>n</math>, and [[abundant number|abundant]] if this sum exceeds <math>n</math>. | |||
The total number of positive divisors of <math>n</math> is a [[multiplicative function]] <math>d(n)</math>, meaning that when two numbers <math>m</math> and <math>n</math> are [[relatively prime]], then <math>d(mn)=d(m)\times d(n)</math>. For instance, <math>d(42) = 8 = 2 \times 2 \times 2 = d(2) \times d(3) \times d(7)</math>; the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However the number of positive divisors is not a totally multiplicative function: if the two numbers <math>m</math> and <math>n</math> share a common divisor, then it might not be true that <math>d(mn)=d(m)\times d(n)</math>. The sum of the positive divisors of <math>n</math> is another multiplicative function <math>\sigma (n)</math> (e.g. <math>\sigma (42) = 96 = 3 \times 4 \times 8 = \sigma (2) \times \sigma (3) \times \sigma (7) = 1+2+3+6+7+14+21+42</math>). Both of these functions are examples of [[divisor function]]s. | |||
{{anchor|number_of_divisors_formula}} | |||
If the [[prime factorization]] of <math>n</math> is given by | |||
:<math> n = p_1^{\nu_1} \, p_2^{\nu_2} \cdots p_k^{\nu_k} </math> | |||
then the number of positive divisors of <math>n</math> is | |||
:<math> d(n) = (\nu_1 + 1) (\nu_2 + 1) \cdots (\nu_k + 1), </math> | |||
and each of the divisors has the form | |||
:<math> p_1^{\mu_1} \, p_2^{\mu_2} \cdots p_k^{\mu_k} </math> | |||
where <math> 0 \le \mu_i \le \nu_i </math> for each <math>1 \le i \le k.</math> | |||
For every natural <math>n</math>, <math>d(n) < 2 \sqrt{n}</math>. | |||
Also,<ref> | |||
{{cite book | |||
| last = Hardy | |||
| first = G. H. | |||
| authorlink = G. H. Hardy | |||
| coauthors = E. M. Wright | |||
| title = An Introduction to the Theory of Numbers | |||
| publisher = Oxford University Press | |||
| date = April 17, 1980 | |||
| location = | |||
| page = 264 | |||
| url = http://archive.org/stream/AnIntroductionToTheTheoryOfNumbers-4thEd-G.h.HardyE.m.Wright#page/n279/mode/2up | |||
| doi = | |||
| id = | |||
| isbn = 0-19-853171-0}} | |||
</ref> | |||
:<math>d(1)+d(2)+ \cdots +d(n) = n \ln n + (2 \gamma -1) n + O(\sqrt{n}).</math> | |||
where <math> \gamma </math> is [[Euler–Mascheroni constant]]. | |||
One interpretation of this result is that a randomly chosen positive integer ''n'' has an expected | |||
number of divisors of about <math>\ln n</math>. | |||
==In abstract algebra== | |||
Given the definition for which <math>0 \mid 0</math> holds, the relation of divisibility turns the set <math>\mathbb{N}</math> of [[non-negative]] integers into a [[partially ordered set]]: a [[lattice (order)|complete distributive lattice]]. The largest element of this lattice is 0 and the smallest is 1. The meet operation '''∧''' is given by the [[greatest common divisor]] and the join operation '''∨''' by the [[least common multiple]]. This lattice is isomorphic to the [[duality (order theory)|dual]] of the [[lattice of subgroups]] of the infinite [[cyclic group]] [[integer|<math>\mathbb{Z}</math>]]. | |||
==See also== | |||
* [[Arithmetic functions]] | |||
* [[Divisibility rule]] | |||
* [[Divisor function]] | |||
* [[Euclid's algorithm]] | |||
* [[Fraction (mathematics)]] | |||
* [[Table of divisors]] — A table of prime and non-prime divisors for 1–1000 | |||
* [[Table of prime factors]] — A table of prime factors for 1–1000 | |||
==Notes== | |||
{{reflist}} | |||
== References == | |||
* {{cite book|last=Durbin|first=John R.|title=Modern Algebra : an Introduction|year=1992|publisher=Wiley|location=New York|isbn=0-471-51001-7|url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000258.html|edition=3rd}} | |||
* [[Richard K. Guy]], ''Unsolved Problems in Number Theory'' (3rd ed), [[Springer Verlag]], 2004 ISBN 0-387-20860-7; section B. | |||
* {{citation|first=I.N.| last=Herstein|title=Abstract Algebra|year=1986|publisher=Macmillan Publishing Company|place=New York|isbn=0-02-353820-1}} | |||
* [[Øystein Ore]], Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints). | |||
* {{citation|first=Charles C.|last=Sims|title=Abstract Algebra: A Computational Approach|year=1984|publisher=John Wiley & Sons|place=New York|isbn=0-471-09846-9}} | |||
{{Divisor classes}} | |||
[[Category:Elementary number theory]] | |||
[[Category:Division]] |
Revision as of 14:48, 3 February 2014
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29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.
In mathematics a divisor of an integer , also called a factor of , is an integer that can be multiplied by some other integer to produce .
Definition
Two versions of the definition of a divisor are commonplace:
- For integers and , it is said that divides , is a divisor of , or is a multiple of , and this is written as
- if there exists an integer such that .[1] Under this definition, the statement holds.
- As before, but with the additional constraint .[2] Under this definition, the statement does not hold.
In the remainder of this article, which definition is applied is indicated where this is significant.
General
Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned.
1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself.[3] Every integer is a divisor of 0.[4] Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd.
1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a trivial divisor is known as a non-trivial divisor. A non-zero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors.
There are divisibility rules which allow one to recognize certain divisors of a number from the number's digits.
The generalization can be said to be the concept of divisibility in any integral domain.
Examples
- 7 is a divisor of 42 because , so we can say . It can also be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42.
- The non-trivial divisors of 6 are 2, −2, 3, −3.
- The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
- , because .
- The set of all positive divisors of 60, , partially ordered by divisibility, has the Hasse diagram:
Further notions and facts
There are some elementary rules:
- If and , then . This is the transitive relation.
- If and , then or .
- If and , then holds, as does .[5] However, if and , then does not always hold (e.g. and but 5 does not divide 6).
If , and gcd, then . This is called Euclid's lemma.
If is a prime number and then or .
A positive divisor of which is different from is called a proper divisor or an aliquot part of . A number that does not evenly divide but leaves a remainder is called an aliquant part of .
An integer whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself.
Any positive divisor of is a product of prime divisors of raised to some power. This is a consequence of the fundamental theorem of arithmetic.
A number is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisors is less than , and abundant if this sum exceeds .
The total number of positive divisors of is a multiplicative function , meaning that when two numbers and are relatively prime, then . For instance, ; the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However the number of positive divisors is not a totally multiplicative function: if the two numbers and share a common divisor, then it might not be true that . The sum of the positive divisors of is another multiplicative function (e.g. ). Both of these functions are examples of divisor functions.
<number_of_divisors_formula>...</number_of_divisors_formula>
If the prime factorization of is given by
then the number of positive divisors of is
and each of the divisors has the form
Also,[6]
where is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer n has an expected number of divisors of about .
In abstract algebra
Given the definition for which holds, the relation of divisibility turns the set of non-negative integers into a partially ordered set: a complete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. The meet operation ∧ is given by the greatest common divisor and the join operation ∨ by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group .
See also
- Arithmetic functions
- Divisibility rule
- Divisor function
- Euclid's algorithm
- Fraction (mathematics)
- Table of divisors — A table of prime and non-prime divisors for 1–1000
- Table of prime factors — A table of prime factors for 1–1000
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7; section B.
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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
- ↑ for instance, Template:Harvnb or Template:Harvnb
- ↑ Template:Harvnb
- ↑ This statement either requires 0|0 or needs to be restricted to nonzero integers.
- ↑ This statement either requires 0|0 or needs to be restricted to nonzero integers.
- ↑ . Similarly,
- ↑
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534